Simplify and expand the following expression: $ \dfrac{p}{2p + 10}+\dfrac{p + 2}{p - 2} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2p + 10)(p - 2)$ Multiply the first term by $\dfrac{p - 2}{p - 2}$ $ \begin{align*} \dfrac{p}{2p + 10} \times \dfrac{p - 2}{p - 2} & = \dfrac{(p)(p - 2)}{(2p + 10)(p - 2)} \\ & = \dfrac{p^2 - 2p}{(2p + 10)(p - 2)}\end{align*} $ Multiply the second term by $\dfrac{2p + 10}{2p + 10}$ $ \begin{align*} \dfrac{p + 2}{p - 2} \times \dfrac{2p + 10}{2p + 10} & = \dfrac{(p + 2)(2p + 10)}{(p - 2)(2p + 10)} \\ & = \dfrac{2p^2 + 14p + 20}{(p - 2)(2p + 10)}\end{align*} $ Now we have: $ = \dfrac{p^2 - 2p}{(2p + 10)(p - 2)} + \dfrac{2p^2 + 14p + 20}{(p - 2)(2p + 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{p^2 - 2p + 2p^2 + 14p + 20}{(2p + 10)(p - 2)} $ $ = \dfrac{3p^2 + 12p + 20}{(2p + 10)(p - 2)}$ Expand the denominator: $ = \dfrac{3p^2 + 12p + 20}{2p^2 + 6p - 20}$